On the eigenvalues of truncations of random unitary matrices

We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and R\'effy identified the limiting spectral measure if $\frac{m}{n}\to\alpha$, as $n\to\infty$; under suitable scaling, the family $\{\mu_\alpha\}_{\alpha\in(0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $\alpha$) and uniform measure on the unit circle (as $\alpha\to1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $\mu_\alpha$ is typically of order $\sqrt{\frac{\log(m)}{m}}$ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new"Coulomb transport inequality"due to Chafa\"i, Hardy, and Ma\"ida.


Introduction
Let U be a random n × n unitary matrix; that is, a random element of U (n) whose distribution is given by Haar measure. By a truncation of such a matrix, we mean a reduction to the upper-left m × m block, for some m ≤ n. In the case that m = o( √ n), the truncated matrix is close to a matrix of independent, identically distributed Gaussian random variables (see Jiang [4]); the circular law for the Ginibre ensemble would lead one to expect that the eigenvalue distribution was approximately uniform in a disc, and this was indeed verified by Jiang in [4]. At the opposite extreme, namely m = n, we have the full original matrix U . The eigenvalues of U itself are also well-understood; it was first proved by Diaconis and Shahshahani [2] that for a sequence {U n } with U n distributed according to Haar measure on U (n), the corresponding sequence of empirical spectral measures converges to the uniform measure on the circle, weakly in probability. In more recent work [5] of the second author and M. Meckes, it was shown that if µ n denotes the spectral measure of U and ν is the uniform measure on the circle, then with probability one, W 1 (µ n , ν) ≤ C √ log(n) n , demonstrating a stronger uniformity of the eigenvalues of such a matrix than, for example, a collection of n i.i.d. uniform points on the circle (whose empirical measure typically has distance of the order 1 √ n from the uniform measure). It is thus natural to consider the evolution of the distribution of the eigenvalues of an m × m trucation U m of U , as α = m n ranges from o(1) to 1 − o(1). In fact, the exact eigenvalue distribution of such truncations is known (see [7]; also [6]) and has density on † Supported in part by NSF DMS 1612589. 1 {|z| < 1} n given by Making use of the explicit eigenvalue density, Petz and Réffy [6] identified the large-n limiting spectral measure of U m , when m n → α ∈ (0, 1); it has radial density with respect to Lebesgue measure on C (as it must, by rotation-invariance), given by While the mathematical motivation in studying the eigenvalues of these truncations, and particularly the evolution of the ensemble as the ratio m n ranges from o(1) to 1 − o(1), is clear, there are also many phyical systems in which large unitary matrices play a central role, and in which truncations of those matrices arise naturally. E.g., in chaotic scattering, the amplitudes of waves coming into the system are related to the amplitudes of outgoing waves by a large unitary matrix (called an S-matrix), and the so-called transmission matrix (related to long-lived resonances of the system) is a truncation of the S-matrix. See, e.g., [3], where the use of random unitary matrices in this context was explored.
The purpose of this paper is to give non-asymptotic results; i.e., to describe the ensemble of eigenvalues of truncations of U ∈ U (n) for fixed (large) n. Our main result on approximation of the spectral measure is the following. Theorem 1. Let P n,m be the joint law of the eigenvalues x 1 , · · · , x m ofŨ m = n m U m . Let µ m be the empirical spectral measure given bŷ Let α = m n , and suppose that C log(n) where C α = min 1 log(α −1 ) , 1 and C ′ α = max log α −1 , 1 .
The bounds in Theorem 1 are tight enough that we can in fact treat the evolution of the process of spectral measures of truncations of U , as the truncation ratio α ranges from o(1) to 1 − o(1). Note that if α = (log n) β n , then r 2 m = 1 a + 1 (log n) 2 −β log log n (log n) β which tends to zero with n only if β > 2. This is the reason for the range of m specified in Theorem 2.
The support of the eigenvalue distribution of the renormalizedŨ m is the disc of radius n m , although the limiting spectral measure ofŨ m is supported on the unit disc; the following treats the question of how far into this intermediate regime the eigenvalues are likely to stray.
This estimate requires some effort to parse. Firstly, observe that choosing ǫ so that Note that in the non-trivial case that α(1 + ǫ) 2 < 1, While the bound stated in Theorem 3 is formally stronger, we will use the simpler bound resulting from the above estimate in the following discussion, separated into three distinct regimes.
(i) α → 0: For α small, our required lower bound on (1 + ǫ) 2 is , then the bound in (2) tends to zero at least as quickly as n −2 , and so it follows from the Borel-Cantelli lemma that with probability one, for any δ > 0, n large enough, and α → 0 as n → ∞, the support of the empirical spectral measure ofŨ m lies within the disc of radius 2 + δ − log(α), as opposed to the a priori support of the disc of radius 1 √ α . (ii) moderate α: Here the bound in (2) tends to zero exponentially with n, and in this case the requirement on ǫ results in a disc of fixed radius r α (somewhat smaller than 1 √ α but still larger than one), so that the empirical spectral measure ofŨ m is supported within the disc of radius r α with probability one, for n large enough. (iii) α → 1: The bound in (2) tends to zero exponentially with n, and for α tending to one, and so if α → 1, then applying the Borel-Cantelli lemma gives that for any ǫ > 0, with probability one, for n large enough, the empirical spectral measure ofŨ m is contained within a disc of radius 1 + ǫ.
1.1. Background. Let U ∈ U (n) be a random matrix whose distribution is given by Haar measure, and let U m denote the m × m principle submatrix of U obtained by removing all but the first m rows and m columns of U . The specific form of the eigenvalue density for U m given in Equation (1) above means that the eigenvalues of U m can be viewed as the (random) locations of m unit charges in a two-dimensional Coulomb gas with external potential. Specifically, if the energy H n,m (z 1 , . . . , z m ) is defined by with the potential V n,m (z) defined by otherwise, then the Gibbs measure on C m (taking the inverse temperature β to be 2) is where λ denotes Lebesgue measure on C. That is, the Gibbs measure in this Coulomb gas model is exactly the eigenvalue density for U m , and so the empirical measure of the charges z 1 , . . . , z m has the same distribution as the empirical spectral measure of U m . This was the viewpoint taken by Petz and Réffy in [6] to identify the large-n limiting spectral measure of U m ; the limiting measure described above is exactly the equilibrium measure for the 2-dimensional Coulomb gas model with potential V α , where m n → α as n → ∞. It should be noted that the viewpoint here is a step removed from the usual Coulomb gas model, where the potential would not depend on m; allowing such a dependence is made possible by the fact that our results are non-asymptotic but hold for fixed n and m.
In recent work, Chafaï, Hardy and Maïda [1] have developed an approach to studying the non-asymptotic behavior of Coulomb gases, using new inequalities they call Coulomb transport inequalities. Specifically, if E(µ) = g(x − y)dµ(x)dµ(y) is the Coulomb energy, with there is a constant C D > 0 such that for any pair of probability measures µ and ν supported on D with E(µ), E(ν) < ∞, When comparing to the equilibrium measure µ V of the Coulomb gas model, this leads to the estimate where E V is the modified energy functional The estimate (3) is the key ingredient in the proof of Theorem 1. The proof follows the analysis in [1] closely, although their results do not apply directly to our potential.
Since our interest is in the evolution of the eigenvalue distribution as m n ranges from essentially zero to 1, it is appropriate to work not with U m itself, but withŨ m = n m U m , so that for any choice of m, the support of the limiting eigenvalue density is the unit disc. By a change of variables, the eigenvalue distribution ofŨ m is given by |z| ≥ n m . Let α := m n ∈ (0, 1). In this normalization, the equilibrium measure is the rotation-invariant probability measure µ α on the unit disc whose density with respect to Lebesgue measure is given by otherwise.

Proofs of the Main Results
We begin with Theorem 3.
Proof of Theorem 3. The form of the eigenvalue density (1), specifically the presence of the Vandermonde determinant, gives that eigenvalues of U m form a determinantal point process on C with the kernel where Θ(x) = ½ (0,∞) (x), and the normalization factor N j is given by Let B r denote the ball of radius r. Then the expected number of eigenvalues of U m outside B √ α(1+ǫ) is given by The sum on the right can be computed using the hockey stick identity: Then Stirling's formula gives that, for m = αn, and so by Markov's inequality, if z 1 , . . . , z m are the eigenvalues of U m , then The central idea of the proof of Theorem 1 is the following simple application of the bound (3). Let z = (z 1 , . . . , z m ), and letμ(z) := 1 m m j=1 δ z j . Given r > 0, Of course, since the measuresμ(z) are singular, the approximate inequality above is invalid, and so part of the argument is to mollify the empirical measures under consideration. Since our potentialṼ is only finite on |z| < 1 √ α , this requires in particular that the probability of any eigenvalues lying to close to the boundary of this disc is small, which follows from Theorem 3. In fact, even further truncation is useful in order to obtain improved control on the constants. Beyond that, all that is really needed is to give estimates for the normalizing constant and the modified Coulomb energy at the equilibrium measure.
The following lemma relates the energyH n,m (x 1 , . . . , x m ) to the modified Coulomb energy of the mollified spectral measure. x :=μ x * λ ǫ , where λ ǫ is the uniform probability measure on the ball B ǫ . Then for x 1 , . . . , Proof. Lemma 4.2 from [1] gives that so that the only task is to give an upper bound for Ṽ * λ ǫ −Ṽ (x i ).
In the proof of Theorem 1, we will use the following version of the Coulomb transport inequality from [1], which is an immediate consequence of Lemma 3.1 together with Theorem 1.1 of that paper.
Lemma 5 (Coulomb Transport Inequality [1]). If D ⊂ R d is compact then for any µ ∈ P(R d ) supported in D, where if R > 0 is such that D ∪ supp(µ α ) ⊂ B R , then C D∪supp(µα) can be taken to be vol(B 4R ).
The probability that the eigenvalues (λ 1 , . . . , λ m ) of n m U m lie in the set A η,r is given by by Lemma 4. The normalizing constantc n,m can be bounded in terms of µ α as follows: where the inequality is by Jensen's inequality.
It follows that and so combining the estimate in (4) with the analysis above gives that P n,m (A η,r ) ≤ exp − m 2 16π(1 + η + ǫ) 2 The claimed result thus follows from the Borel-Cantelli lemma.